Investigating the Koch Snowflake

Mathematics HL Portfolio

Omar Nahhas.

Class 12 “IB” (C).

The Koch snowflake is also known as the , which was first described by Helge von Koch in 1904. Its building starts with an , removing the inner third of each side, building another  with no base at the location where the side was removed, and then repeating the process indefinitely.

The first three stages are illustrated in the figure below

Each step in the process is the repeating of the previous step hence it is called iteration.

If we let Nn = the number of sides, ln = the length of a single side, Pn = the length of the perimeter, and An = the area of the snowflake, all at nth stage, we shall get the following table for the 1st three iterations.

Table no.1: the value of Nn, ln, Pn, and An, at the stage zero and the following three stages.

Note: assume that the initial side length is 1.

We can see from the above table that the number of sides isn multiplied by four at each iteration. The length of each side is divided by 3 in each step, thus it is 1/3 the length of the same side in the previous step. As for the perimeter, the perimeter equals the number of sides multiplied by the length of a single side, hence that we have the above values where that

Pn = Nn × ln. As for the area of the diagram, it is equal to the area of the original triangle plus the area of the new smaller triangles added in each step, and since it’s an equilateral triangle its height (the original triangle) is equal to (√3)/2, which was found from the fact that the angels of the equilateral triangle are 60o, and using a segment which bisected the base and was normal to it. So using known famous triangles which the angles 30 o, 60 o, and 90o

** Each graph of every single set of values plotted against the ...